Lesson 2.1: Quadratic Functions


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1 Quadratic Functions: Lesson 2.1: Quadratic Functions Standard form (vertex form) of a quadratic function: Vertex: (h, k) Algebraically: *Use completing the square to convert a quadratic equation into standard quadratic function form. *Use evaluation and solve for the missing value of a when given a vertex and a point. How does the vertex of a parabola relate to the Maximum or Minimum of a quadratic function? Is it possible to find the vertex of a parabola written in general form without completing the square? For each function find the (a)vertex (b) x & y intercepts (c) zeros (d) sketch it.
2 Write the following parabola in vertex form: Vertex (3, 2) passes through (5,7) Vertex (1,2); passes through (4,4) #62, #64 True or False: Without using your calculator, do the graphs of and have the same axis of symmetry.
3 Lesson 2.2: Polynomial Functions of Higher Degree *Graphs of polynomial functions are continuous, meaning it has no breaks. *Graphs of polynomial functions only have smooth rounded turns as opposed to sharp pointed turns. Polynomial function: When n is odd: END BEHAVIOR: A graph with a positive leading coefficient falls to the left and rises to the right. Left endbehavior: Right endbehavior: A graph with a negative leading coefficient rises to the left and falls to the right. Left endbehavior: Right endbehavior: When n is even: A graph with a positive leading coefficient rises to the left and right. Left endbehavior: Right endbehavior: A graph with a negative leading coefficient falls to the left and right. Left endbehavior: Right endbehavior: The Intermediate Value theorem: Let a and b be real numbers such that a < b. If f is a polynomial function such that f(a) f(b), then in the interval [a,b], f takes on every value between f(a) and f(b).
4 Polynomial function: The graph of f has at most n real zeros. The function f has at most (n 1) relative extrema. If f is a polynomial function and a is a real number: 1. x = a is a zero of the function f. 2. x = a is a solution of the polynomial equation f(x) = (x a) is a factor of the polynomial f(x). 4. (a, 0) is an xintercept of the graph of f. In general, a factor of yields a repeated zero x = a of multiplicity k: If k is odd, the graph crosses the xaxis at x = a If k is even, the graph touches (but does not cross) the xaxis at x = a Find all the real zeros of the polynomial functions Find a polynomial function that has the given zeros. 2, 6 0,2,5 Can you solve these equations? From which polynomial (simplest) did we come from?
5 Analyze the given functions providing the x&y intercepts, zeros, action at the zeros, endbehavior. Then use your calculator to find the relative maximums & minimums, and give the increasing & decreasing intervals. #86, #88
6 Lesson 2.3: Real Zeros of Polynomial Functions Division Algorithm: Dividend = Divisor*Quotient + Remainder Long Division: Synthetic Division (when the divisor is in the form of x k) The Remainder Theorem: If a polynomial f(x) is divided by x k, the remainder is r = f(k) Use the remainder theorem to find the remainder. The Factor Theorem: A polynomial f(x) has a factor (x k) if and only if f(k) = 0 Is (x1) a factor of the following function? f (x) = 2x 3 + 3x 2 8x + 3 The Rational Zero Test: see Pg. 118 List the possible rational zeros of f, use a calculator to narrow the list down, find the zeros. Given one real zero of a function find the others. ; x = ½
7 Lesson 2.4: Complex Numbers Complex number in standard form: a + bi, for real numbers a and b Conjugate: 4+6i 6 10i Complex plane: Simplify:
8 Lesson 2.5: The Fundamental Theorem of Algebra Linear Factorization Theorem: If f(x) is a polynomial of degree n, where n > 0, f has precisely n linear factors where are complex numbers and is the leading coefficient of f(x) If a + bi, where b 0, is a zero of a function, then the conjugate a bi is also a zero. Find all the zeros of the function and write the polynomial as a product of linear factors. Find the polynomial function with integer coefficients that has the given zeros. 2, 4 + i, 4 i 2,2,2,4i, 4i Use the given zero to find all the zeros of the function. ; zero: r = 2i ; zero:
9 Lesson 2.6: Rational Functions and Asymptotes Rational Function:, p(x) & q(x) are polynomials and q(x) isn t the zero polynomial. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either ; where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has at most one horizontal asymptote, as follows a. If n < m, the xaxis (y = 0) is a horizontal asymptote b. If n = m, the line is a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote Analyzing graphs of rational functions 1.The yint (if any) is the value of f(0) 2. The xints (if any) are the zeros of the numerator 3. The vert. asy.(if any) are the zeros of the denominator 4. The horizontal asy.: see a,b,and c above * It s all about the degree! Find the domain, and identify any horizontal and vertical asymptotes
10 f (x) = x 5 x Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. g(x) = x 2 4 x + 3 h(x) = x 2 +1 Write a rational function f having the specified characteristics. (There are many correct answers.) Vertical asymptotes: x = 2, x = 1 Vertical asymptotes: x = 0, x = 2.5 x = 5 Horizontal asymptotes: y = 3
11 Lesson 2.6: Rational Functions and Asymptotes Rational Function:, p(x) & q(x) are polynomials and q(x) isn t the zero polynomial. Horizontal Asymptote The line y = b is a horizontal asymptote of the graph of a function y = f(x) if either Vertical Asymptote The line x = a is a vertical asymptote of the graph of a function y = f(x) if either ; where p(x) and q(x) have no common factors. 1. The graph of f has vertical asymptotes at the zeros of q(x) 2. The graph of f has at most one horizontal asymptote, as follows a. If n < m, the xaxis (y = 0) is a horizontal asymptote b. If n = m, the line is a horizontal asymptote c. If n > m, the graph of f has no horizontal asymptote Analyzing graphs of rational functions 1.The yint (if any) is the value of f(0) 2. The xints (if any) are the zeros of the numerator 3. The vert. asy.(if any) are the zeros of the denominator 4. The horizontal asy.: see a,b,and c above * It s all about the degree! Find the domain, and identify any horizontal and vertical asymptotes
12 f (x) = x 5 x Find the zeros (if any) of the rational function. Use a graphing utility to verify your answer. g(x) = x 2 4 x + 3 h(x) = x 2 +1 Write a rational function f having the specified characteristics. (There are many correct answers.) Vertical asymptotes: x = 2, x = 1 Vertical asymptotes: x = 0, x = 2.5 x = 5 Horizontal asymptotes: y = 3
13 Lesson 2.7: Graphs of rational functions If the degree of the numerator is exactly 1 more than the degree of the denominator, the graph of the rational function has a slant asymptote *use division to identify the slant asymptote. Find the intercepts and asymptotes to help you sketch the given rational functions.
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